Finite Group Databasebeta

← #17045455

Subgroups — #17045455

Order 4352.

1308 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762943C211yesmaximal, characteristic
2C4×#10587616C211yesmaximal, characteristic, complemented
3#14529649C212yesmaximal, complemented
4#14529470C212yesmaximal, complemented
5#14529401C212yesmaximal, complemented
6#14529135C212yesmaximal, complemented
7#14529101C211yesmaximal, characteristic, complemented
8#14528972C211yesmaximal, characteristic, complemented
9#14528715C211yesmaximal, characteristic, complemented
10C4×C22⋊#3622C211yesmaximal, characteristic, complemented
11C2×C17×#390C211yesmaximal, characteristic, complemented
12C22×#63228C2212yes—
13C2×#10587616C2211yescharacteristic
14C2×C4×D68C2211yescharacteristic, complemented
15C2×C4×C22⋊D17—21no—
16#11762943C418yescomplemented
17C2×#10587478C2212yescomplemented
18C2×#10587503C2212yes—
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×#10587480C2212yescomplemented
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×C4×#3622C2212yescomplemented
25#11762711C2211yescharacteristic, complemented
26#11762707C2211yescharacteristic, complemented
27#11762700—21no—
28#11762688—21no—
29#11762669C2211yescharacteristic
30C4×C4⋊Dic17C2211yescharacteristic, complemented
31C4×#63227—21no—
32#11762571C2212yescomplemented
33#11762571C2212yescomplemented
34#11762564C2212yescomplemented
35C23×D4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17—21no—
38C22×C22⋊C4×C17—21no—
39C22×C42×C17C2211yescharacteristic
40C17×#390—21no—
41C17×#390—21no—
42C17×#390C2212yes—
43C17×#390C2212yes—
44C25×C17—21no—
45C22×D4×C17—21no—
46C22×D4×C17—22no—
47C22×D4×C17—42no—
48C22×D4×C17—42no—
49C22×D4×C17C2×C414yescomplemented
50C22×D4×C17C2312yes—
51C23×C4×C17—21no—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17D411yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C22×C22⋊D17—21no—
56C22×D68C2311yescharacteristic
57C22×C4×D17—21no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—42no—
60C2×C4⋊C4×C17C2312yes—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—22no—
65C2×C22⋊C4×C17—44no—
66C2×C22⋊C4×C17—44no—
67C2×C42×C17—42no—
68C2×C42×C17D411yescharacteristic
69C2×C42×C17D411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×#63228—42no—
74C2×#63228—42no—
75C2×#63228C2×C418yescomplemented
76#10587616—22no—
77#10587616C2×C414yes—
78C4×C22⋊D17—44no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×#63225—21no—
87C2×C4×Dic17—21no—
88C22×#3622C2312yes—
89C22×#3622C2312yes—
90C4×D68—42no—
91#10587503—42no—
92#10587503—42no—
93#10587503C2×C418yescomplemented
94#10587502—21no—
95#10587502—21no—
96#10587502—21no—
97C22⋊#3622—22no—
98C22⋊#3622C2×C414yescomplemented
99#10587480—42no—
100#10587478—42no—
101C4×#3622—42no—
102C24×C17—24no—
103C24×C17—22no—
104C24×C17—42no—
105C24×C17—42no—
106C24×C17—42no—
107C24×C17—42no—
108C24×C17—42no—
109C24×C17—42no—
110C24×C17C2×D411yescharacteristic
111C23×D17—21no—
112C2×D4×C17—24no—
113C2×D4×C17—22no—
114C2×D4×C17—22no—
115C2×D4×C17—22no—
116C2×D4×C17—22no—
117C2×D4×C17—22no—
118C2×D4×C17—48no—
119C2×D4×C17—42no—
120C2×D4×C17—42no—
121C2×D4×C17—48no—
122C2×D4×C17C22×C414yes—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—41no—
129C22×C4×C17—41no—
130C22×C4×C17—41no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17D814yescomplemented
135C22×C4×C17C22×C412yes—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17#4112yes—
142C22×C4×C17C2411yescharacteristic
143C2×C22⋊D17—24no—
144C2×C22⋊D17—42no—
145C2×C22⋊D17—42no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C2×D68—22no—
150C2×D68—22no—
151C2×D68—21no—
152C2×D68C22×C412yes—
153C2×C4×D17—42no—
154C2×C4×D17—42no—
155C2×C4×D17—42no—
156C4⋊C4×C17—22no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—42no—
159C22⋊C4×C17—44no—
160C22⋊C4×C17—44no—
161C22⋊C4×C17—44no—
162C22⋊C4×C17—44no—
163C42×C17—22no—
164C42×C17—22no—
165C42×C17—41no—
166C42×C17—41no—
167C22⋊Dic17—42no—
168#63228—48no—
169#63227—24no—
170#63227—42no—
171#63227—42no—
172#63227—42no—
173C4⋊Dic17—21no—
174C4⋊Dic17—22no—
175C4⋊Dic17—22no—
176C4⋊Dic17C22×C412yes—
177#63225—42no—
178C4×Dic17—42no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622C22×C414yes—
185#22040—171nomaximal, Sylow
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—24no—
191C23×C17—48no—
192C23×C17—48no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—48no—
200C23×C17—84no—
201C23×C17—84no—
202C23×C17—84no—
203C23×C17—84no—
204C23×C17C4×D412yes—
205C23×C17C2×D812yes—
206C23×C17#13512yes—
207C23×C17C22×D411yescharacteristic
208C22×D17—22no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215D4×C17—24no—
216D4×C17—24no—
217D4×C17—48no—
218D4×C17—42no—
219D4×C17—42no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—44no—
234C2×C4×C17—44no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—84no—
239C2×C4×C17C4×D412yes—
240C2×C4×C17C2×D812yes—
241C2×C4×C17C2×D812yes—
242C2×C4×C17AGL(1,ℤ/8)12yes—
243C2×C4×C17C23×C411yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17H(5,𝔽2)12yes—
246C2×C4×C17H(5,𝔽2)12yes—
247C22⋊D17—48no—
248C2×Dic17—22no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257D68—24no—
258D68—42no—
259C4×D17—84no—
260#3622—24no—
261#3622—42no—
262C2×#447—171no—
263C4×#121—171no—
264C2×#390D1711yescharacteristic, complemented
265#1970—172no—
266#1934—172no—
267#1925—172no—
268#1879—172no—
269#1874—171no—
270#1852—171no—
271#1786—171no—
272C4×C22⋊C8—171no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—44no—
280C22×C17—44no—
281C22×C17—44no—
282C22×C17—816no—
283C22×C17—82no—
284C22×C17—82no—
285C22×C17—82no—
286C22×C17—82no—
287C22×C17C4×D814yes—
288C22×C17C2×C4×D411yescharacteristic
289C22×C17#53411yescharacteristic
290C22×C17#53511yescharacteristic
291C22×C17C22×D811yescharacteristic
292C22×C17C2×#13511yescharacteristic
293C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C17#57511yescharacteristic
295C2×D17—42no—
296C2×D17—42no—
297C2×D17—42no—
298C2×D17—81no—
299C2×D17—81no—
300C2×D17—81no—
301C2×D17—81no—
302C4×C17—22no—
303C4×C17—22no—
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—41no—
307C4×C17—82no—
308C4×C17#51812yes—
309C4×C17#53812yes—
310Dic17—42no—
311Dic17—81no—
312Dic17—81no—
313C23×D4C2×D1711yescharacteristic
314C22×D8—172no—
315C2×#121—171no—
316C2×C4×D4—171no—
317C2×C4×D4—341no—
318C22×C4⋊C4C2×D1711yescharacteristic
319C22×C22⋊C4—21no—
320C22×C22⋊C4—21no—
321C22×C42C2×D1711yescharacteristic
322#447—178no—
323C2×#107—172no—
324C2×#102—172no—
325C2×#102—172no—
326C2×#102—172no—
327C2×#102—172no—
328C2×C22⋊C8—171no—
329C2×C22⋊C8—171no—
330C2×C4×C8—172no—
331#390—21no—
332#390—21no—
333#390—171no—
334#390C2×D1712yes—
335#390C2×D1712yes—
336#389—171no—
337#388—341no—
338#386—341no—
339#380—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—341no—
342#340—172no—
343#340—172no—
344#336—172no—
345C2×C17—24no—
346C2×C17—84no—
347C2×C17#232111yescharacteristic
348C2×C17C2×C4×D811yescharacteristic
349C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
350C2×C17#318611yescharacteristic
351C2×C17#318811yescharacteristic
352C2×C17#320511yescharacteristic
353C2×C17#321111yescharacteristic
354D17—82no—
355C25—21no—
356C22×D4—22no—
357C22×D4—21no—
358C22×D4—42no—
359C22×D4—42no—
360C22×D4—171no—
361C22×D4—341no—
362C22×D4C4×D1714yescomplemented
363C22×D4C22×D1712yes—
364C23×C4—21no—
365C23×C4—341no—
366C23×C4C22⋊D1711yescharacteristic
367C23×C4C22⋊D1711yescharacteristic
368C23×C4C22×D1711yescharacteristic
369C2×D8—178no—
370C2×D8—682no—
371C2×D8—682no—
372C22×C8—172no—
373C22×C8—172no—
374#121—174no—
375#121—342no—
376C4×D4—682no—
377C4×D4—684no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4C22×D1712yes—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—44no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C42—42no—
397C2×C42—341no—
398C2×C42C22⋊D1711yescharacteristic
399C2×C42C22⋊D1711yescharacteristic
400C2×C42C22×D1711yescharacteristic
401C2×C42C22×D1711yescharacteristic
402#107—682no—
403#102—178no—
404#102—682no—
405#102—682no—
406#102—682no—
407C22⋊C8—174no—
408C22⋊C8—342no—
409C4×C8—682no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2204011yescharacteristic, complemented, Sylow
414C24—22no—
415C24—24no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—341no—
423C24C2×C22⋊D1711yescharacteristic
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—48no—
434C2×D4—172no—
435C2×D4—344no—
436C2×D4—341no—
437C2×D4—342no—
438C2×D4—342no—
439C2×D4—682no—
440C2×D4—682no—
441C2×D4C2×C4×D1714yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4#6322814yescomplemented
460C22×C4C2×C4×D1712yes—
461C22×C4#6325212yes—
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468D8—688no—
469C2×C8—174no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8—342no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—172no—
479C4⋊C4—341no—
480C4⋊C4—342no—
481C4⋊C4—342no—
482C4⋊C4—682no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—344no—
488C22⋊C4—682no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—682no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—342no—
516C23—681no—
517C23—681no—
518C23—681no—
519C23—681no—
520C23—681no—
521C23—681no—
522C23C4×C22⋊D1712yes—
523C23C2×#6322812yes—
524C23#1058763612yes—
525C23C22×C22⋊D1711yescharacteristic
526D4—24no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—344no—
532D4—688no—
533D4—682no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—342no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—1364no—
563C2×C4C4×C22⋊D1712yes—
564C2×C4C2×#6322812yes—
565C2×C4C2×#6322812yes—
566C2×C4#1058761812yes—
567C2×C4C22×C4×D1711yescharacteristic
568C2×C4C2×#6325211yescharacteristic
569C2×C4#1058769112yes—
570C2×C4#1058769112yes—
571C8—344no—
572C8—682no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—682no—
588C22—682no—
589C22—682no—
590C22—1361no—
591C22—1361no—
592C22—1361no—
593C22—1361no—
594C22C4×#6322814yes—
595C22C2×C4×C22⋊D1711yescharacteristic
596C22#1176336811yescharacteristic
597C22#1176337611yescharacteristic
598C22C22×#6322811yescharacteristic
599C22C2×#1058763611yescharacteristic
600C22C2×#1058761811yescharacteristic
601C22#1176360711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—82no—
608C4—682no—
609C4—1361no—
610C4—1361no—
611C4#1176328212yes—
612C4#1176339712yes—
613C2—24no—
614C2—84no—
615C2—1362no—
616C2#1453144211yescharacteristic
617C2C2×C4×#6322811yescharacteristic
618C2C4×#1058761811yescharacteristic
619C2#1453953211yescharacteristic
620C2#1453954911yescharacteristic
621C2#1453970211yescharacteristic
622C2#1453974911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#390C2
socleC23×C17C22×D4
prime radicalC2×#11762943C2
cosocleC22×C4×C17C24
hypercenterC2×#390D17
hypocenterC17#22040
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4×#10587616C2semidirect
3. #14529649C2semidirect
4. #14529470C2semidirect
5. #14529401C2semidirect
6. #14529135C2semidirect
7. #14529101C2semidirect
8. #14528972C2semidirect
9. #14528715C2semidirect
10. C4×C22⋊#3622C2semidirect
11. C2×C17×#390C2semidirect
14. C2×C4×D68C22semidirect
16. #11762943C4semidirect
17. C2×#10587478C22semidirect
21. C2×#10587480C22semidirect
24. C2×C4×#3622C22semidirect
25. #11762711C22semidirect
26. #11762707C22semidirect
30. C4×C4⋊Dic17C22semidirect
32. #11762571C22semidirect
33. #11762571C22semidirect
34. #11762564C22semidirect
49. C22×D4×C17C2×C4semidirect
75. C2×#63228C2×C4semidirect
93. #10587503C2×C4semidirect
98. C22⋊#3622C2×C4semidirect
134. C22×C4×C17D8semidirect
264. C2×#390D17semidirect
362. C22×D4C4×D17semidirect
413. C17#22040semidirect
459. C22×C4#63228semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117629432
C4×#105876162
#145296492
#145294702
#145294012
#145291352
#145291012
#145289722
#145287152
C4×C22⋊#36222
C2×C17×#3902
#2204017